Ashoke DAS, Ashis BİSWAS, Bappaditya DEBNATH

Riemann Solutions (LCS)n-Manifolds Admitting Different Semi-Symmetric Structures

The object of the present paper is to study the Riemannian solitons on (LCS)nmanifolds and we observed in this case the Riemann soliton on M is shrinking, steady or expanding according to α 2−ρ being positive, zero or negative respectively. Here also we discussed the Riemann solitons in (LCS)n-manifold admitting (i) R·C = 0, R·K = 0, (ii) E ·C = 0, E ·K = 0, (iii) R·R = 0, R·P = 0, R·E = 0, R·P ∗ = 0, R·M = 0, R ·Wi =0, R ·W∗ i = 0, (iv) E ·R = 0, E ·P = 0, E · E = 0, E ·P ∗ = 0, E ·M = 0, E · Wi = 0 and E · W∗ i = 0.( for all i = 1, 2, ....9). We found that the Riemann soliton on M is shrinking, steady or expanding according to the conditions (i) α 2 − ρ being positive, zero or negative respectively, (ii) [k(n−1) (n − 2) (1+α 2−ρ)−kr−r] being positive, zero or negative respectively and (iv) α 2 − ρ being negative, zero or positive. But for the condition (iii) the Riemann soliton on M is always steady.

Keywords:

Reimannian Solution, (LCS)n-manifold, semi-symetric structure,

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[1] R. S. Hamilton, ”The Ricci flow on surfaces, Mathematics and general relativity,” Contemp. Math. American Math. Soc., vol. 71, pp. 237-262, 1988.
[2] Udriste, C., ”Riemann flow and Riemann wave,” Ann. Univ. Vest, Timisoara. Ser. Mat.-Inf., vol. 48(1-2), pp. 265-274, 2010.
[3] Udriste, C., ”Riemann flow and Riemann wave via bialternate product Riemann- ian metric,” preprint, arXiv.org/math.DG/1112.4279v4, 2012.
[4] Hirica, I.E. and Udriste, C., ”Ricci and Riemann solitons,” Balkan J. Geom. Applications, vol. 21, no. 2, pp. 35-44, 2016.
[5] Devaraja N., Aruna K. H. & Venkatesha V., ”Riemann soliton within the framework of contact geometry”, Quaestiones Mathematicae, vol. 1-15, 2020.
[6] Venkatesha V., Aruna K. H. & Devaraja N., ”Riemann solitons and almost Riemann solitons on almost Kenmotsu manifolds,” International Journal of Geometric Methods in Modern Physics, DOI: 10.1142/S0219887820501054, 2020.
[7] Stepanov S. E. and Tsyganok I. ”The theory of infinitesimal harmonic transformations and its applications to the global geometry of Riemann solitons,” Balkan Journal of Geometry and Its Applications, vol. 24, no. 1, pp. 113-121, 2019.
[8] Szabo, Z. I., ”Structure theorems on Riemannian spaces satisfying R(X, Y ) ´ ·R = 0, I (the local version),” J. Diff. Geom., vol. 17, pp. 531-582, 1982.
[9] Deshmukh, Sharief & De, Uday & Zhao, Peibiao, ”Ricci semisymmetric almost Kenmotsu manifolds with nullity distributions,” Filomat., vol. 32, pp. 179-186. 10.2298/FIL1801179D, 2018.
[10] De, U.C & Han, Y. & Mandal, K., ”On Para-Sasakian manifolds satisfying certain curvature conditions,” Filomat. vol. 31, pp. 1941-1947, 10.2298/FIL1707941D, 2017.
[11] Mondal A. & De U. C., ”Quarter-Symmetric Nonmetric Connection on P -Sasakian Manifolds”, ISRN Geometry, 10.5402/2012/659430, 2012.
[12] Bagewadi, C. S. and Venkatesha, ”Some curvature tensors on a Trans-Sasakian manifold,” Turk J Math., vol. 31, pp. 111-121, 2007.
[13] Ozgur, C. and M. M. Tripathi, ”On P-Sasakianmanifolds satisfying certain conditions on the concircular curvature ¨ tensor” Turk. J. Math., vol. 31, pp. 171-179, 2007.
[14] Szabo, Z. I., ”Classification and construction of complete hypersurfaces satisfying R(X, Y ) ´ ·R = 0,” Acta.Sci. Math., vol. 7, pp. 321-348, 1984.
[15] A. A. Saikh and H. Kundu, ”On equivalency of various geometric structures,” J. Geom, DOI 10.1007/s00022-013- 0200-4.
[16] Yano, K. and Bochner, S., ”Curvature and Betti numbers”, Annals of Mathematics Studies, Princeton University Press, vol. 32, 1953.
[17] Eisenhart, L. P., ”Riemannian Geometry”, Princeton University Press, 1949.
[18] Ishii, Y., ”On conharmonic transformations”, Tensor (N.S.), vol. 7, pp. 73-80, 1957.
[19] Pokhariyal, G. P. and Mishra R.S., ”Curvature tensor and their relativistic significance II”, Yokohama Math. J. vol. 19, no. 2, pp. 97-103, 1971.
[20] Pokhariyal, G. P. ”Relativistic significance of curvature tensors”, Int.J.Math. Math. Sci. vol. 5, no 1, pp. 133-139, 1982.
[21] Pokhariyal, G. P. and Mishra R.S., Curvature tensor and their relativistic significance, Yokohama Math. J. vol. 18, no 2, pp. 105-108, 1970.

Cankaya University Journal of Science and Engineering-Cover

Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2009
Yayıncı: Çankaya Üniversitesi

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